On the regularity of three-dimensional rotating euler-boussinesq equations

The 3-D rotating Boussinesq equations (the "primitive" equations of geophysical fluid flows) are analyzed in the asymptotic limit of strong stable stratification. The resolution of resonances and a nonstandard small divisor problem are the basis for error estimates for such fast singular oscillating limits. Existence on infinite time intervals of regular solutions to the viscous 3-D "primitive" equations is proven for initial data in H^α, α ≥ 3/4. Existence on a long-time interval T* of regular solutions to the 3-D inviscid equations is proven for initial data in H^α, α > 5/2 (T* → ∞ as the frequency of gravity waves → ∞).